A unified periodical boundary conditions for representative volume elements of composites and applications

Abstract

An explicit unified form of boundary conditions for a periodic representative volume element (RVE) is presented which satisfies the periodicity conditions, and is suitable for any combination of multiaxial loads. Starting from a simple 2-D example, we demonstrate that the “homogeneous boundary conditions” are not only over-constrained but they may also violate the boundary traction periodicity conditions. Subsequently, the proposed method is applied to: (a) the simultaneous prediction of nine elastic constants of a unidirectional laminate by applying multiaxial loads to a cubic unit cell model; (b) the prediction of in-plane elastic moduli for [±θ]_n angle-ply laminates. To facilitate the analysis, a meso/micro rhombohedral RVE model has been developed for the [±θ]_n angle-ply laminates. The results obtained are in good agreement with the available theoretical and experimental results.

Introduction

Composite materials are becoming an essential part of present engineered materials because they offer advantages such as higher specific stiffness and strength, better fatigue strength and improved corrosion-resistance compared to conventional materials. They are used in various applications ranging from aerospace structures to sports equipment, electronic packaging, medical tools, and civil engineering structures. Consequently, prediction of the mechanical properties of the composites has been an active research area for several decades. Except for the experimental studies, either micro- or macromechanical methods are used to obtain the overall properties of composites.

Micromechanical method provides overall behavior of the composites from known properties of their constituents (fiber and matrix) through an analysis of a periodic representative volume element (RVE) or a unit-cell model (Aboudi, 1991; Nemat-Nasser and Hori, 1993). In the macromechanical approach, on the other hand, the heterogeneous structure of the composite is replaced by a homogeneous medium with anisotropic properties. The advantage of the micromechanical approach is not only the global properties of the composites but also various mechanisms such as damage initiation and propagation, can be studied through the analysis (Xia et al., 2000; Ellyin et al., 2002).

There are several micromechanical methods used for the analysis and prediction of the overall behavior of composite materials. In particular, upper and lower bounds for elastic moduli have been derived using energy variational principles, and closed-form analytical expressions have been obtained (Hashin and Shtrikman, 1963; Hashin and Rosen, 1964). Based on an energy balance approach with the aid of elasticity theory, Whitney and Riley (1966) obtained closed-form analytical expressions for a composite’s elastic moduli. Unfortunately, the generalization of this method to viscoelastic, elastoplastic and nonlinear composites is very difficult. Aboudi (1991) has developed a unified micromechanical theory based on the study of interacting periodic cells, and it was used to predict the overall behavior of composite materials both for elastic and inelastic constituents. In his work, homogeneous boundary conditions were applied to the RVE or unit cell models. In fact, this is only valid for those cases in which normal tractions are applied on the boundaries. For a shear loading case, many researchers, e.g., Needleman and Tvergaard (1993), Sun and Vaidya (1996), Suquet (1987), among others, have indicated that the ‘plane-remains-plane’ boundary conditions are over-constrained boundary conditions. In the current paper we shall further demonstrate that they are not only over-constrained boundary conditions but may also violate the stress/strain periodicity conditions.

The above micromechanical models can be regarded as mechanical or engineering models. A mathematical counterpart to such engineering methods appeared in the 1970s under the general heading of the ‘asymptotic homogenization theory’. The fundamentals of this theory can be found, e.g. in Suquet (1987), Benssousan et al. (1978), Sanchez-Palencia (1980), and Bakhvalov and Panasenko (1984), among others. Asymptotic homogenization theory has explicitly used periodic boundary conditions in modeling of linear and nonlinear composite materials. These results have clearly shown that characteristic modes of deformation do not result in plane boundaries after deformation (Suquet, 1987). Guedes and Kikuchi (1991) discussed the application of finite element method (FEM) to composite problems. Recent applications of homogenization theory for various aspects of composite analysis are given, for instance, in Raghavan et al. (2001) and Moorthy and Ghosh (1998).

Hori and Nemat-Nasser (1999) presented a universal inequalities which indicate that the predicted effective elastic modulus can vary depending on the applied conditions on the boundary ∂V of a unit cell, and the homogeneous displacement and homogeneous traction boundary conditions will give the upper and lower bounds of the effective modulus. Hollister and Kikuchi (1992) have given a very good comparison of the homogenization theory and the mechanical methods (it is called average field theory in Hori and Nemat-Nasser (1999)), concluding that the homogenization theory, which uses the periodic boundary conditions, yields more accurate results. It is shown that the homogenization theory and mechanical methods can be related to each other and a more applicable hybrid theory was established (Hollister and Kikuchi, 1992).

FEM has been extensively used in the literature to analyze a periodic unit cell, to determine the mechanical properties and damage mechanisms of composites (Adams and Crane, 1984; Aboudi, 1990; Allen and Boyd, 1993; Bonora et al., 1994; Pindera and Aboudi, 1998). In most cases, the applications are limited to the unidirectional laminates. A few investigators have also applied the micromechanical analysis to the cross-ply laminates (laminates contain only 0° and 90° laminae), for which the thermal residual stresses, crack initiation and propagation, viscoplastic or viscoelastic behaviors have been studied (Xia et al., 2000; Ellyin et al., 2002; Bigelow, 1993; Chen et al., 2001).

In the present paper the FEM micromechanical analysis method is applied to unidirectional and angle-ply laminates subject to multiaxial loading conditions. For the latter laminates, special meso/micro rhombohedral RVE models have been developed. Based on general periodicity conditions stated by Suquet (1987), an explicit form of boundary conditions suitable for FEM analyses of parallelepiped RVE models subjected to multiaxial loads is presented. Starting from a simple 2-D example, the results of the present method and those obtained by applying homogeneous boundary conditions are compared. Subsequently, the FEM analyses are conducted for two composite RVE models: (1) a unidirectional laminate to predict simultaneously all nine elastic constants by applying multiaxial loads; (2) a thick [±θ]_n angle-ply laminate to predict simultaneously the four in-plane elastic moduli by applying biaxial loads. The predicted properties are compared with available theoretical or experimental results and are found to be in very good agreement. Although the illustrative analyses presented in the current paper are limited to the elastic range, the basic relations proposed in this paper are independent of the properties of the constituents of the composite.